New Relations Between Discrete and Continuous Transition Operators on (Metric) Graphs

Lenz, Daniel; Pankrashkin, Konstantin

doi:10.1007/s00020-015-2253-2

Cited by 8 publications

(6 citation statements)

References 38 publications

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“…Actually, far more than (2.36) is known in the case of equilateral quantum graphs. In fact, there is a sort of unitary equivalence between equilateral quantum graphs and the corresponding combinatorial Laplacians (see [53,54] and also [46]).…”

Section: Corollary 23 ([26]mentioning

confidence: 99%

Spectral estimates for infinite quantum graphs

Kostenko

Nicolussi

2018

Calc. Var.

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We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.2010 Mathematics Subject Classification. Primary 34B45; Secondary 35P15; 81Q35.

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Section: Corollary 23 ([26]mentioning

confidence: 99%

Spectral estimates for infinite quantum graphs

Kostenko

Nicolussi

2018

Calc. Var.

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“…It is well-known that D −1 A is the transition matrix associated with a random walk on the graph and it has been shown in [CW07] that an interesting interplay exists between the spectral properties of D −1 A and those of a certain bounded, self-adjoint operator A over L 2 (G), where G is the quantum graph associated with G. Such an A can be interpreted as the operator that maps functions over a quantum graph G into their average over balls of radius 1 with respect to the canonical structure of a quantum graph as a metric measure space. Spectral relations between D −1 A and A as well as a representation of A in terms of the quantum graph Laplacian via self-adjoint functional calculus have been proved in [CW07,LP16]. The considerations in [CW07] suggest that the correct quantum graph counterpart of our (unnormalized!)…”

Section: Backward Evolution Equation On General Line Graphsmentioning

confidence: 87%

Dynamical systems associated with adjacency matrices

Mugnolo¹

2018

Discrete &Amp; Continuous Dynamical Systems - B

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We develop the theory of linear evolution equations associated with the adjacency matrix of a graph, focusing in particular on infinite graphs of two kinds: uniformly locally finite graphs as well as locally finite line graphs. We discuss in detail qualitative properties of solutions to these problems by quadratic form methods. We distinguish between backward and forward evolution equations: the latter have typical features of diffusive processes, but cannot be wellposed on graphs with unbounded degree. On the contrary, well-posedness of backward equations is a typical feature of line graphs. We suggest how to detect even cycles and/or couples of odd cycles on graphs by studying backward equations for the adjacency matrix on their line graph.

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“…A reduction principle connecting the eigenvalues of the Kirchhoff Laplacian on a finite metric graph to those of the normalized discrete Laplacian on the underlying combinatorial graph was first observed by von Below in [Bel85]. His results were extended to the different parts of the spectrum of more general operators on possibly infinite graphs, beginning with [Cat97]; we refer to [Pan12,LP16] and references therein for later refinements of Cattaneo's results. In Section 6 we prove that this spectral correspondence between combinatorial and metric graphs carries over to general Schrödinger operators and can be extended to general L p -spaces and certain spaces of continuous functions on the one hand and specific discrete spaces.…”

Section: Introductionmentioning

confidence: 86%

Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and p-independence of spectra

Becker,

Gregorio,

Mugnolo

2020

Preprint

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We analyze properties of semigroups generated by Schrödinger operators −∆ + V or polyharmonic operators −(−∆) m , on metric graphs both on L p -spaces and spaces of continuous functions. In the case of spatially constant potentials, we provide a semi-explicit formula for their kernel. Under an additional sub-exponential growth condition on the graph, we prove analyticity, ultracontractivity, and pointwise kernel estimates for these semigroups; we also show that their generators' spectra coincide on all relevant function spaces and present a Kreȋn-type dimension reduction, showing that their spectral values are determined by the spectra of generalized discrete Laplacians acting on various spaces of functions supported on combinatorial graphs.

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New Relations Between Discrete and Continuous Transition Operators on (Metric) Graphs

Cited by 8 publications

References 38 publications

Spectral estimates for infinite quantum graphs

Spectral estimates for infinite quantum graphs

Dynamical systems associated with adjacency matrices

Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and p-independence of spectra

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